The Plane Triangle and Its Six Circles

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—i8q—

curves, according to the relation which we choose to establish between
x and y. The same is true of all other complex functions.

To recapitulate: The introduction of the imaginary numbers into
Analytical Geometry enables us to construct all functions of the abscis-
sa, whether real or imaginary. For example the equations

y 2 -f- x 2 + 4J — 2x -j- 8 = o,

y + 3y 2 + •** — 5* + IO = °> & c -' & c -j

without the introduction of imaginary numbers cannot be constructed;
but when expressing the condition that y is to be perpendicular upon x,
it is easy to construct them. x\nd indeed they must each represent
some curve since y varies when x does.

THE PLANE TRIANGLE AND ITS SIX CIRCLES.

BY ASHER B. EVANS, A. M., LOCKPORT, N. Y.

The six circles whose properties are discussed in this article are the
circumscribed, the inscribed, the nine-point, and the three escribed circles.
The first two of these circles are familiar to every student of elementary
geometry. The nine-point circle in a triangle is that circle whose cir-
cumference passes through the feet of the three perpendiculars from the
angles upon the opposite sides, the three middle points of the sides, and
the three middle points of the segments of the perpendiculars between
the angles and their common point of meeting. The escribed circles are
three circles situated wholly without the triangle, each of which is tan-
gent to one side of the triangle and to the other two sides produced.

Three points beirg in general sufficient to determine a circumference,
it is necessary to show that the nine points enumerated in the definition
of the nine-point circle are always on the same circumference. To this
end let ABC (Fig. 1) be a triangle, O' the centre of its circumscribed
circle, O'a, O'/?, O'y the perpendiculars from O' to the sides BC, AC,
AB, respectively. Produce O'a to A', 0'/3 to B', O'y to C, making
«A' = O'a, /?B' = O'/J, rC = O'y; complete the triangle A'B'C, let
L be the centre of its circumscribed circle, and let a', j3' t f be the inter-
sections of AL, BL, CL with B'C, A'C, A'B'.

i<po-

i. The triangles ABC, A'B'C are
lequal and their corresponding sides are
■parallel.

For y being the middle point of AB and
[O'C, and j9 the middle point of AC and
lO'B', the three lines BC, yfi, C'B' are par-
lallel and BC = 2^9 = C'B'. Similarly
|AC = 2ya = C'A' and AB = 2/9<z = B'A'.

2. The triangles a/3?-, a! fly 1 are equal and
Itheir corresponding sides are parallel.

Since BO' = BC = C'A = C'L = L'B.
a is the middle point of C'B'. Similarly /?' and y' are the middle points
of C'A' and A'B'.

3. L and O' are the intersections of perpendiculars of the triangles
ABC and A'B'C

For AL, BL, CL being perpendicular to C'B', C'A', A'B', are per-
pendicular to BC, AC, AB (1); and A'O', B'O', CO' being perpendic-
ular to BC, AC, AB are perpendicular to C'B', C'A', A'B'.

4. The triangles ABC, A'B'C are reciprocal.

For O', L and a, a' being homologous points in these two triangles.
A'O' -- AL, aO' = ah; and therefore A'« = A«' and Aa' = a'L'
Similarly B? = j8'L and Cy' = r 'L.

5. If a circumference be described with M the middle point of LO'
as centre and ^O'C as radius, it will circumscribe the triangles afty, a'ft'y'
and pass through f, g f r, the feet of the perpendiculars of the triangle
ABC.

For the similar triangles 0'M«, O'LA' give t ±Jjf = ±±L,

«M

= *A'L

iO'C.

Similarly /9M = |0'C and r M = $0'C. Since ««\
/9/3', yy' are diagonals of the parallelograms «L«'0', /3L/3'0', yLy'O',
they are bisected in M bv the common diagonal O'L . • . «M == a'M
= /?M = /9'M = j-M = y'M = |0'C. Moreover L being the intersec-
tion of perpendiculars (3), and aa', ftp', yy' diameters of the circumfer-
ence circumscribing afty and a'fi'y', this circumference passes through f.

*The engraver has made several mistakes in lettering Fig. 1. For
d and d' read a and a', for B and B' within the triangle read fl and /?',
for // read f, for T and T' read y and y', and for D read L. — Ed.

— rgi —

q and r, the feet of the perpendiculars, and is therefore the circumfer-
ence of the nine-point circle of the triangle ABC.

6. The circumference of the nine-
point circle bisects the straight line join-
ing any point in the circumference of I
the circumscribing circle and the inter- [
section of the perpendiculars.

For E (Fig. 2) being any point on the I
circumference of the circumscribing cir-
cle, and MF being drawn parallel to I
O'E and cutting EL in F,
O'E LE LO' ., ,.

MF = LF = LM =2 - ••• theradlus

of the nine-point circle being JO'E, F the middle point of LE, is on the
circumference of the nine-point circle.

7. If the perpendiculars Ap, Bq, O, produced, meet the circumfer-
ence of the circumscribing circle in A', B', C, the lines LA', LB', LC
will be bisected by BC, AC, AB, respectively.

For the circumference of the nine-point circle passes through f, q, r,
and bisects LA', LB', LC (6).

8. If the diameter AO'P be drawn, PL and BC will bisect each
other.

For D being the middle point of PL, O'D is parallel to AL and hence
perpendicular to BC. Moreover the foot of the perpendicular from O'
upon BC and the middie point of PL are both upon the circumference
of the nine-point circle; therefore D is on BC and the truth of the the-
orem is evident.

9. Four circles may be described, each of which shall touch the
three sides of a plane triangle or those sides produced. If six straight
lines be drawn joining the centres of these circles, two and two, prove
that the middle points of these six lines are in the circumference of a
circle circumscribing the given triangle.

(The figure can readily be supplied by the student.)
Let ABC be the triangle; O, O t , 2 , O s , the centres of the inscribed
and three escribed circles. Then since O x A, 2 B, O s C are the per-
pendiculars of the triangle OjC^Oj and O their intersection, the circle
circumscribing ABC is the nine-point circle of the triangle O x 2 3 ;

— 192 — ■

therefore the circumference ol this circle bisects the six lines OO t , 00 2 ,
OO,, O^, 2 3 , 0,0 3 .

10. R and r being the radii and O' and O the centres of the circum-
scribed and inscribed circles, and L the intersection of perpendiculars,

(O'L) 2 — 2(OL) 3 = R 2 — 4 ;».

Since (Fig. 3) /O'AL = O'AB — LAB = (B — C) and /OAL

= OAB — LAB = 1(B — C) the triangles O'AL, OAL give

(O'L) 2 = (AO') 2 + (AL) 2 — 2AO'.ALcos(B — C) and

(OL) 2 = (AO) 2 + (AL) 2 — 2AO.ALcosi(B — C); or, since r

= AOsiniA, AL = 2RcosA(AL=A'0'=20'«=2RcosA as is seen in
Fig. 1) and r = 4 Rsin|Asin^Bsin^C (Chauvenet's Trig., Eq. 298),

(O'L) 2 = R 2 — 4 R 2 [cos(B — C) — cosA]cosA = R 2

— 8R 2 cosAcosBcosC and

= d£* + ** WA - ^o^-in|AcosKB - C)]
= 2t* + 4 R 2 cos'A + ^^ { 1- ^ sini Acosl(B- C) }

= 2r 2 — 4R 2 cosA(sinBsinC — cos A)

= 2r* — 4R 2 cosAcosBcosC

. • . (O'L) 2 — 2(OL) 2 = R» — 4 r*.

11. R and r a being the radii and O and O x the centres of the cir-
cumscribed and one of the escribed circles, and L the intersection of
perpendiculars (O'L) 2 — 2(O x L) 2 - R 2 — V \.

Since (Fig. 3) /O a AL = ZOAL = ±(B — C), we have from the
triangle C^AL, (O a L) 2 = (AO a ) 2 + ( KLf — 2A0 1 .ALcos 2 -(B - C);
or, since r 2 = AO^in^A, A L = 2RcosA, and r =

4 RsinjAcoslBcosJC (Chauvenet's Trig., Eq. 298),

O lL ) 2 = -JL + 4 R 2 cos 2 A - iROiL^^B-C)
1 ' sin 2 -|A ^ sin^A

ir

+ 4 R 2 cos 2 A — l^J 00 ^ [2sin^Acos|(B - C)]

1 — cos A ' 1 — cosA

*93-

= 2rf — 4R ? cosA(sinBsinC - cosA)
= 2r\ — 4R 2 cosAcosBcosC. Since by (n)

(O'L) 2 = R 2 — 8R 2 cosAcosBcosC, we have
(O'L) 2 — 2(0^)* = R* — \r\.

12. The nine-point circle is tangent
to the inscribed circle internally.

The centre of the nine-point circle
being at M (Fig. 3) the middle point of
LO', the triangle LOO' gives

2(OL) 2 + 2(00') 2 = 4(OM) 2 + (O'L) 2 .

Since (10)
(O'L 2 — 2(OL) 3 = R 2 - 4^ and
(OO') 2 = R 2 - 2Rr (Chauvenet's Trig ,

Eq. 300),
4 (0 M) 2 = R 2 — 4 Rr + 4^, and
. • . OM = iR — r.

The radius of the nine-point circle being iR, the truth of the propo-
sition is evident.

13. The nine-point circle is tangent to each of the three escribed
circles externally-

In this case the triangle LOjO' (Fig. 3) gives

2(0,L) 2 + 2(0! 0')» = ^OjM J* + (O'L) 2 .

Since (11) (O'L) 2 — 2(OjL) 2 == R 2 — v \ and (O, O'f = R 2 + 2RH
(Chauvenet's Trig., Eq. 301),

4 (0 ,M) J = R 2 + 4 Rr, + 4 rf, and

.-. OjM = |R + r x . Similarly, r v r 3 being the radii
and 2 , 3 the centres of the other two escribed circles,

2 M = {R + r 2
and 3 M=|R4-r 3

The distances between the centre of the nine-point circle and the cen-
tres of the escribed circles show that the nine-point circle is tangent to
each of the escribed cireles and moreover that the contact is external.

14. The nine-point circle of the triangle ABC (Fig. 1) is tangent to

- ip4 -

the thirty-two inscribed and escribed circles of the triangles ABC,
ALC, BLC, ALB, A'B'C, A'O'C, B'O'C, A'O'B'.

Since a, /3, y, a', /3', y', the middle points of the sides of these eight tri-
angles, are on the circumference of the nine-point circle of the triangle
ABC, this nine-point circle is common to the eight triangles, and it is
therefore tangent to their eight inscribed and twenty-four escribed cir-
cles.

15. The intersection of perpendiculars, the centre of the nine-point
circle, the centre of gravity of the triangle, and the centre of the cir-
cumscribed circle, are four points in the same straight line, and their
distances apart are in a eonstant ratio.

Let these four points be designated by L, M, G, O' (Fig. 1). As A«.
B;?, Cy intersect in the centre of gravity of the triangle ABC, G is the
centre of similitude of ABC and a$y. It is evident that M, L are the
centres of similitude of a'fi'y' and afiy, ABC and a'fi'y, respectively .
Since G, M, L are the three centres of similitude of the three triangles
ABC, afly, a'ft'y' whose sides are respectively parallel, these three points
are on the same straight line (Hackley's Geometry, Appendix 2, p. 2):
but LM passes through O'; therefore the four points G, M, L, O r are
colinear.

L and O' being the intersections of perpendiculars, and G the centre
of similitude of the triangles ABC, ajiy, we have

LG: GO'::AB:a/9::2 : 1.

Therefore, since M is the middle point of LO',

LM __ £ MG __ 1 G(y 1
LO 7- V LO r ~6' LO' "3"

The nine-point circle is celebrated in the history of geometry*. Some
of its properties were given by a German geometrician as early as 1822.
Steiner gave the property of contact with the inscribed and escribed
circles in 1828 in Gergonne's Annates. The designation " nine-point "
was first given to this circle by Terquam in 1842, in the Nouvelles An-
nates des Math. This interesting circle began to attract the attention o)
English mathematicians about twenty years ago, and its properties have
been given in various prize questions proposed at the English Universi-
ties; but I am not aware that any American mathematician has treated
of its most simple properties.